The interaction of electron beams with periodic gratings enables the generation of coherent radiation over a wide spectral range in synchrotrons and free-electron lasers [1]}, [2]}. Indeed, far-field radiative components are produced in general upon scattering of the evanescent electromagnetic field that accompanies electrons in motion when they cross or pass near a material structure, giving rise to cathodoluminescence emission [3]}. If the structure is periodically patterned with spatial period \(d\)  along the electron velocity vector \({\bf v}\) , Smith-Purcell (SP) radiation [4]} is emitted over a broad range of wavelengths \(\lambda \)  and angles \(\theta _n\)  relative to \({\bf v}\)  satisfying the condition of far-field constructive interference
\(c/v-\cos \theta _n=n\lambda /d, \) 
